Herodotos states the surface
of the faces is equal to the square of the height and it is plethra.

If,

s = semiside

a = apothema

h = height

Herodotos reports:

h²= as

By theorem of Pythagoras

a² = s² x h²

Hence,

a² = s² x as

a² = s(a x s)

s: a =(a x s)

This means that the apothema
and the semiside are in relation of golden section Herodotos computes
the surface by plethra. By plethron he refers to the square with a side
of 100 Egyptian royal cubits. This is the Egyptian acre, that is the amount
plowed in a day.

The Egyptian acre, called
st‘’t, has a side of 100 cubits and a surface of 2756 square meters;
ít is similar to the Roman iugerum of 2524 square meters. In other
parts of his work Herodotos colls this unit by the name of aroura, which
is the term used in documents of Hellenistic Egypt; in Roman times Latin
documents of Egypt use the term iugerum.

If the height of the Pyramid
is 280 cubits, the surface would be 78,400 square cubits, and not 80,000.

The reason for this is that
agrarian units were arranged in a series in which each one is double of
the preceding one, Each succeeding one is conceived as constructed on
the diagonal; the relation between the side and the diagonal is calculated
use by the simple relation 5;7. For instance, the double aroura is conceived
as a square with a side of 140 cubits, instead of 141,421. But the surface
of the quadruple aroura is conceived as constructed on the diagonal of
the double aroura, using the relation 7;10 between side and diagonal,
so that the quadruple aroura comes at correctly as a square with a side
of 200 cubits.

This the aroura come surface
19,600 square conversely, the half aroura is conceived as a square with
side of 70 cubits, but the quarter of aroura is a square with a side
of 50 cubits.

By this procedure the double
aroura comes to have a surface of 140² = 19,600 square cubits, instead
of 20,000.

The double aroura so calculated
is 49/50 of the exact figure, this approximation was take into account
by assuming that the aroura had a side of 99 cubits instead of 100. A
square with side of 99 cubits has a surface of 9801 square cubits which
con be considered the exact half of a square with a side of 140 cubits
(surface of 19,600 square cubits).

Herodotos must have followed
a calculation which assumes an aroura with side of 99 cubits.

The method of calculation
is made clear by the Pomponius Mela from which we get that the square
of the height and the surface of the foces is 4 iugera.

By iugerum Mela means
a double aroura with sides of 140 cubits. This way of reckoning is more
simple since the height of the Pyramid is 280 cubits, If is immediately
evident that if the height is 280 cubits, the square of the height is
4 double arourai.

But by exact reckoning the
surface is something less from 4 double aroura; hence Mela says quattuorfereiugera, ”almost four acres’!

Herodotos must have followed
the same way of computing, except that he counted by single arourai
with side of 99 cubits, arriving at the figure of 8 arourai.

It must be help in mind that
the division according to the Golden Section was practically important
in the triplication and quintuplication of squares. It is significant
in the thirteenth Book of Eudid the Golden Section is introduced in relation
to the triplication and quintuplication of squares (Proposition 1-6).

Implication and quintuplication
of squares was necessary when units of surface were arronged according
to the sexagesimal system. Implication is necessary for onedecimal reckoning
and quintuplication for decimal reckoning.

If the side of a basic square
is computed as “the part” in a Golden Section, by addling” the rest”
twice to it, there is obtained the side of a square treble in surface.
If “the rest” is added twice to the whole segment divided by the Golden
Section, there rents the side of a square quintuple in surface. In other
works,

3 may be computed as

1 x 2/ (1- 1/) and 5 as

1 x 2/ or / x 1/.

For approximations it was
to 99/100 of the side of square and to add of this lenght to of

All this could calculated
quite simple in practical reckoning, by using a square with side 99 instead
of 100. If one tokes the side of the basic square as 99 and adds to it
3/4 of this length, obtains a side of 173,250, which is the side of a
treble square (3 =1,73205), If one takes the side of the basic square
which is 100 and adds to it 5/4 of 99, he obtains a length of 223.750,
which is the side of a square quintuple in surface (5 = 2.23607).

This kind of reckoning may
explain why the surface of the foces of the Great Pyramid is calculated
by arourai with side of 99 and why the relation between the side and the
apothema is 5;4 when the pyramidion is not included in the reckoning.